## Airy-Gauss beams and their transformation by paraxial optical systems

Optics Express, Vol. 15, Issue 25, pp. 16719-16728 (2007)

http://dx.doi.org/10.1364/OE.15.016719

Acrobat PDF (660 KB)

### Abstract

We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation.

© 2007 Optical Society of America

## 1. Introduction

1. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE , **54**, 1312–1329 (1966). [CrossRef]

*k*is the wave number in the propagation medium, and (

*x, z*) are the transverse and longitudinal coordinates, respectively. The most familiar solutions to Eq. (1) are the Hermite-Gaussian beams, whose mathematical and physical properties are well known [2]. Three-dimensional solutions of the (2+1)D PWE can be readily constructed with products of planar solutions, i.e.

*U*(

*x,y, z*)=

*U*(

_{x}*x, z*)

*U*(

_{y}*y, z*).

3. E. G. Kalnins and W. Miller Jr., “Lie theory and separation of variables. 5. The equations iUt +Uxx=0 and iUt+Uxx-c/x2 U=0,” J. Math. Phys. **15**, 1728–1737 (1974). [CrossRef]

4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264–267 (1979). [CrossRef]

*z*=0 is assumed to have the form

*U*(

*x*,0)=Ai(

*bx*), where Ai denotes the Airy function and

*b*is a real and positive constant. The Airy beams lack parity symmetry about the origin, propagate in free space without distortion, and exhibit a peculiar parabolic transverse shift with propagation distance. The existence of this lateral shift motivated Berry and Balazs to adopt the term

*accelerating*for describing the Airy wave packets. Some mathematical and physical properties of the Airy wave packets were further explored by Besieris et al. [5

5. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. **62**, 519–521 (1994). [CrossRef]

6. K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy Packet in quantum mechanics,” Am. J. Phys. **64**, 1034–1036 (1996). [CrossRef]

7. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

*finite-energy Airy beam*whose distribution at

*z*=0 is given by

*U*(

*x*,0)=Ai(

*bx*)exp(

*ax*), with a being a positive real quantity in order to ensure square integrability of the beam. By employing the Fourier decomposition of plane waves, Siviloglou and Christodoulides derived a closed-form expression for the propagation of the finite-energy Airy beams in free space, and also showed that these beams still exhibit the distinctive parabolic lateral shift with propagation distance. More recently, Besieris and Shaarawi [8

8. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

*z*axis, and the beam variance (i.e. the second-order moment) increases quadratically with propagation distance. The first experimental generation of Airy beams was reported recently by Siviloglou et al. in Ref. [9

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**, 213901 (2007). [CrossRef]

4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264–267 (1979). [CrossRef]

7. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

## 2. Airy-Gaussian beams through ABCD systems

*x*

_{1},

*z*

_{1}) of a first-order optical ABCD system as

*κ*

_{1},

*δ*

_{1},

*S*

_{1}, and

*q*

_{1}are complex quantities in the most general situation. The scaling parameter

*κ*

_{1}controls the spatial frequency of the transverse field oscillations. The parameter

*S*

_{1}allows for the possibility that the input field

*U*

_{1}has a transverse decay and tilt. The complex beam parameter

*q*

_{1}provides an initial Gaussian apodization and spherical wavefront [2]. The parameter

*δ*

_{1}introduces an optional lateral shift in the Airy function and in the linear phase term, but it does not affect the Gaussian modulation. The overall amplitude factor exp (

*iS*

^{3}

_{1}/3) is included for later convenience. Equation (2) reduces to the finite-energy Airy beams [7

7. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

*δ*

_{1}=0 and

*q*

_{1}=∞, and further to the Airy beams [4

4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264–267 (1979). [CrossRef]

*S*

_{1}=0.

*U*

_{1}(

*x*

_{1}) through the ABCD system can be performed with the Huygens diffraction integral [2]

*U*

_{2}(

*x*

_{2}) is the field at the output plane (

*x*

_{2},

*z*

_{2}). The integration yields

*q*

_{1}travelling axially through the ABCD system, and the transformation laws for the parameters

*q*

_{2},

*κ*

_{2},

*S*

_{2}, and

*δ*

_{2}, from the input plane

*z*

_{1}to the output plane

*z*

_{2}are

*κ*

_{1}and

*κ*

_{2}are not proportional to each other through a real factor leading to different intensity profiles of the function

*U*, and moreover because the parameters

*q*

_{1}and

*κ*

_{1}are transformed according to different laws.

**32**, 979–981 (2007). [CrossRef] [PubMed]

8. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

*δ*

_{1}=0 and

*q*

_{1}=∞, namely

**47**, 264–267 (1979). [CrossRef]

*S*

_{1}=0 in Eqs. (8).

## 3. Physical discussion

### 3.1. Free space propagation

*U*(

*x, z*) traveling into the half-space

*z*≥ 0. The field at the input plane

*z*=0 is given by Eq. (2) and the ABCD transfer matrix for free space propagation from the plane

*z*

_{1}=0 to the plane

*z*2=

*z*reads as [

*A,B;C,D*]=[1

1. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE , **54**, 1312–1329 (1966). [CrossRef]

*z*;0,1

1. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE , **54**, 1312–1329 (1966). [CrossRef]

*µ=µ*(

*z*)≡1+

*z*/

*q*

_{1}.

*z*, where

_{R}*z*

_{R}=kw^{2}

_{1}/2 is the Rayleigh distance associated to the initial Gaussian modulation of width

*w*

_{1}. The corresponding cross sections of the amplitude and phase profiles at the plane

*z*=0 are included at the left side of the Fig. 1. For all examples the scaling parameter

*κ*

_{1}is assumed real and equal to

*w*

_{1}/6. The field distributions

*U*(

*x, z*) were obtained by calculating Eq. (4) using Eq. (9) at 200 transverse planes evenly spaced from the input to the output plane.

*x*)

*z*employing a Gaussian-Legendre quadrature method.

*q*

_{1}~0) and (

*S*

_{1}/

*kκ*

_{1}) negligible. Under these conditions, the interpretation of the AiG beam as accelerating, i.e., one characterized by a nonlinear lateral shift with range, depends on the

*z*

^{2}term of the displacement parameter

*δ*(

_{2}*z*) in Eq. (9). For small values of |

*κ*

_{1}|, such as those considered in Refs. [7

**32**, 979–981 (2007). [CrossRef] [PubMed]

8. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

*S*

_{1}/

*κ*

_{1}) is the linear decay constant,

*β*=Re(

*S*

_{1}/

*κ*

_{1}) is the tilt transverse wave number,

*R*

_{1}=1/Re(

*q*

^{-1}

_{1}) is the radius of curvature of the initial spherical phase front (

*R*

_{1}<0 convergent phase front,

*R*

_{1}>0 divergent phase front), and

*R*

_{1}=-(0.2

*z*)/3. Since the quadratic phase factor exp (

_{R}*ikx*

^{2}/2

*R*

_{1}) can be interpreted as the transmittance of a lens of focal distance

*f*=|

*R*

_{1}|, Fig. 1(b) represents also the focusing evolution of the initial field in Fig. 1(a). The propagation of the beam centroid exactly follows the geometrical path, as expected from the ray optics approach [2]. The presence of the quadratic phase factor introduces a transverse momentum such that the centroid shifts laterally on propagation following the linear trajectory

*x*(

_{c}*z*)=

*x*(0)(1+

_{c}*z/R*

_{1}), where

*x*(0) is the centroid position at the initial plane

_{c}*z*=0. Note that

*R*

_{1}is negative for a converging initial wavefront, then in this case

*x*(

_{c}*z*)=0 when

*z*=|

*R*

_{1}|.

*x*(0)/

_{c}*R*

_{1}+

*β/k*vanishes, thus the position of the centroid remains constant on propagation. Finally, the effect of the displacement

*δ*

_{1}and an initial converging spherical wavefront (

*R*

_{1}<0) is shown in Fig. 1(e), where we can see that although the initial intensity pattern and its centroid position have changed, the centroid trajectory is still linear and crosses the optical axis at the equivalent focal plane

*z*=|

*R*

_{1}|.

*δ*

_{1}=0 and

*q*

_{1}=∞ can be derived using the symmetry properties of the PWE [Eq. 1]. Specifically, given the fundamental Gaussian solution GB(

*x, z*) to the PWE, then

*H*(

*x, z*)=GB(

*x, z*)

*F*(

*x/µ, z/µ*) is also a solution provided that

*F*(

*x, z*) is a solution to the PWE. Choosing

*F*(

*x, z*) to be the finite energy Airy beam given in Ref. [7

**32**, 979–981 (2007). [CrossRef] [PubMed]

### 3.2. Propagation through a quadratic index medium

*n*(

*r*)=

*n*

_{0}(1-

*x*

^{2}/2

*a*

^{2}). The ABCD transfer matrix from plane

*z*

_{1}=0 to plane

*z*

_{2}=

*z*is given by

*z*is described by Eq. (4). Substitution of the matrix elements in Eq. (13) into Eqs. (6) yields the parameter transformations:

*L*=2

*π a*, therefore, the initial field self-reproduces after a distance

*L*.

*z=L*/4=

*πa*/2, the ABCD matrix Eq. (13) reduces to [0,

*a*;-1/

*a*,0] which is indeed identical to the matrix transformation from the first to the second focal plane of a converging thin lens of focal length

*a*, i.e. a Fourier transformer. From Eqs. (14) we see that at the Fourier plane

*z=L*/4 the beam parameters become

*U*̃

_{1}(

*kx*)=∫∞-∞

*U*

_{1}(

*x*)exp(-

*ik*)d

_{x}x*x*of the AiG beam

*U*

_{1}(x) in Eq. (2) can be determined by inserting Eqs. (15) into Eq. (4) and making the replacements

*x*→

*k*and

_{x}*a*→

*k*. In a similar way, for intermediate planes

*z=pL*/4, the substitution of Eqs. (14) into Eq. (4) yields the

*p*th fractional Fourier transform of the initial field [10].

*L*. As predicted by geometric optics, the centroid oscillates back and forth across the optical axis following a cosine trajectory with a zero initial slope. The combined effect of an intial tilt and a lateral shift

*δ*

_{1}is shown in Fig. 2(b). For this case the initial slope of the centroid trajectory is determined by the angle tilt and then the centroid does not cross the Fourier plane at the optical axis.

## 4. Analysis of the square integrability

*w*) on the complex plane

*w=u+iv*as shown in Fig. 3. As the radius |

*w*| increases, the amplitude of the Airy function |Ai(

*w*)| decreases monotonically in the sector defined by |argw|<

*π*/3, increases indefinitely in the sector

*π*/3≤ |arg

*w*|<

*π*, and oscillates with a decreasing average along the negative real axis arg

*w*=

*π*. Zeros of the Airy function in the complex plane are located on the negative part of the real axis.

*δ*and

*κ*, each value of

*x*∈(-∞,∞) determines a point [

*u*(

*x*),

*v*(

*x*)] on the complex plane

*w=u+iv*=(

*x*+

*δ*)/

*κ*. The parametric equations for

*u*(

*x*) and

*v*(

*x*) are given by

*κ*·

*δ*)≡Re

*κ*Reδ

*+*Im

*κ*Im

*δ*, and (

*κ*×

*δ*)≡Re

*κ*Im

*δ*-Im

*κ*Re

*δ*. By combining

*u*(

*x*) and

*v*(

*x*) to eliminate

*x*, we obtain that the point [

*u*(

*x*),

*v*(

*x*)] falls on the straight line

*U*(

*x*)|

^{2}for large values of |

*x*|. Useful asymptotic representations of the Airy functions for large argument are given by [11, 12]:

*v*(

_{s}*u*) coincides with the real axis

*v*=0. From Eq. (18) we see that this condition is fulfilled when Im

*κ*=0 and Im

*δ*=0, and therefore the argument of the Airy function becomes purely real, i.e.

*w*∈R. Applying Eqs. (19) and (20) we obtain for Re

*κ*>0

*v*(

_{s}*u*) does not coincide with the real axis

*v*=0. Using Eq. (19) we obtain

*w*)=arctan{[-

*x*Im

*κ*+(

*κ*×

*δ*)]/[

*x*Re

*κ*+(

*κ*·

*δ*)]}∈(-

*π,π*), and irrelevant overall constants have been omitted.

*q*

_{1})>0] is sufficient for ensuring the square integrability of the AiG beam independently of the values taken by the other constants. Conversely, the condition Im(1/

*q*

_{1})<0 causes the field to diverge as |

*x*|→∞.

*q*

_{1})=0], then from Eqs. (21)–(23) we conclude that the beam is still square integrable in the following two symmetrical cases: (a)

*κ*∈R+=(0,∞),

*δ*∈R, and Im

*S*<0 (i.e.

*α*>0), (b)

*κ*∈R-=(-∞,0),

*δ*∈R, and Im

*S*>0 (i.e.

*α*<0). The case (a) is indeed the required condition to ensure square integrability of the finite-energy Airy beams introduced in Refs. [7

**32**, 979–981 (2007). [CrossRef] [PubMed]

**32**, 2447–2449 (2007). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE , |

2. | A. E. Siegman, |

3. | E. G. Kalnins and W. Miller Jr., “Lie theory and separation of variables. 5. The equations iUt +Uxx=0 and iUt+Uxx-c/x2 U=0,” J. Math. Phys. |

4. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

5. | I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. |

6. | K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy Packet in quantum mechanics,” Am. J. Phys. |

7. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

8. | I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. |

9. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

10. | H. M. Osaktas, Z. Zalevski, and M. A. Kutay, |

11. | M. Abramowitz and I.A. Stegun, |

12. | O. Vallée and M Soares, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(070.2590) Fourier optics and signal processing : ABCD transforms

(260.1960) Physical optics : Diffraction theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 14, 2007

Revised Manuscript: November 23, 2007

Manuscript Accepted: November 24, 2007

Published: December 3, 2007

**Citation**

Miguel A. Bandres and Julio C. Gutiérrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express **15**, 16719-16728 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16719

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### References

- H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE, 54, 1312-1329 (1966). [CrossRef]
- A. E. Siegman, Lasers (University Science, Mill Valley CA, 1986).
- E. G. Kalnins and W. MillerJr., "Lie theory and separation of variables. 5. The equations iUt +Uxx = 0 and iUt +Uxx-c/x2 U = 0," J. Math. Phys. 15, 1728-1737 (1974). [CrossRef]
- M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979). [CrossRef]
- I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994). [CrossRef]
- K. Unnikrishnan and A. R. P. Rau, "Uniqueness of the Airy Packet in quantum mechanics, " Am. J. Phys. 64, 1034-1036 (1996). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007). [CrossRef] [PubMed]
- I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
- H. M. Osaktas, Z. Zalevski, and M. A. Kutay, The Fractional Fourier Transform with applications in Optics and Signal processing (Wiley, London, 2001).
- M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1964).
- O. Vallée and M Soares, Airy functions and applications to physics (Imperial College Press, London, 2004).

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